Abstract : The classical von Neumann inequality provides a fundamental link between complex analysis and operator theory. It shows that for any contraction $T$ on a Hilbert space and any polynomial $p$, the operator norm of $p(T)$ satisfies
$\|p(T)\| \leq \sup _{|z| \leq 1}|p(z)|$
Whereas Andô extended this inequality to pairs of commuting contractions, the corresponding statement for triples of commuting contractions is false. However, it is still not known whether von Neumann's inequality for triples of commuting contractions holds up to a constant. I will talk about this question and about function theoretic upper bounds for $\|p(T)\|$.

Keywords : von Neumann's inequality; Andô's inequality; commuting tuples of contractions; polydisc; Besov norm